Theorem 19.2OLD 1700

Description: Obsolete version of 19.2 1659 as of 4-Dec-2017. (Contributed by NM, 2-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
19.2OLD	⊢ (∀xφ → ∃xφ)

Proof of Theorem 19.2OLD

Step	Hyp	Ref	Expression
1		equid 1676	. . 3 ⊢ x = x
2	1	notnoti 115	. . . 4 ⊢ ¬ ¬ x = x
3	2	spfalw 1672	. . 3 ⊢ (∀x ¬ x = x → ¬ x = x)
4	1, 3	mt2 170	. 2 ⊢ ¬ ∀x ¬ x = x
5		idd 21	. . 3 ⊢ (x = x → (φ → φ))
6	5	speimfw 1645	. 2 ⊢ (¬ ∀x ¬ x = x → (∀xφ → ∃xφ))
7	4, 6	ax-mp 5	1 ⊢ (∀xφ → ∃xφ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)